{"paper":{"title":"On singular integrals with non-negative kernels in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"L^2 boundedness of the K_4 singular integral on a 1-Ahlfors regular set in the Heisenberg group implies it lies in a 1-Ahlfors regular curve.","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Lingxiao Zhang, Sean Li, Vasileios Chousionis","submitted_at":"2026-05-17T22:41:12Z","abstract_excerpt":"In this paper we revisit nonnegative kernels in the first Heisenberg group $\\He$, and in particular we further study the family $$K_\\alpha(x,y,z)= \\frac{|z|^{\\alpha/2}}{\\|(x,y,z)\\|_{H}^{\\alpha+1}}, \\quad \\alpha>0,$$ which was introduced in \\cite{CL}.\n  We first show that if $E \\subset \\He$ is a $1$-Ahlfors regular set and the SIO associated with the kernel $K_4$ is $L^2(E)$-bounded, then $E$ is contained in a $1$-Ahlfors regular curve. Combined with the converse implication which was obtained by F\\\"assler and Orponen in \\cite{FO1dim}, our result provides a characterization of uniform $1$-recti"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If E subset He is a 1-Ahlfors regular set and the singular integral operator associated with the kernel K_4 is L^2(E)-bounded, then E is contained in a 1-Ahlfors regular curve.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The 1-Ahlfors regularity of the set E together with the non-negativity and precise homogeneity of the kernel K_4 are assumed to control the maximal function and cancellation properties needed for the implication to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"L2-boundedness of the SIO with kernel K_4 on 1-Ahlfors regular sets in the Heisenberg group characterizes containment in 1-Ahlfors regular curves, with negative results for alpha in (0,2) and a bounded operator on a purely 1-unrectifiable set.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"L^2 boundedness of the K_4 singular integral on a 1-Ahlfors regular set in the Heisenberg group implies it lies in a 1-Ahlfors regular curve.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"26686f0fd2a79c7e2c57e7919b18a1c30592647922daf00619976045dcfd0f1a"},"source":{"id":"2605.17680","kind":"arxiv","version":1},"verdict":{"id":"ea6d1be5-9ffa-43df-9bb4-21f91a5bfaeb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:57:23.462304Z","strongest_claim":"If E subset He is a 1-Ahlfors regular set and the singular integral operator associated with the kernel K_4 is L^2(E)-bounded, then E is contained in a 1-Ahlfors regular curve.","one_line_summary":"L2-boundedness of the SIO with kernel K_4 on 1-Ahlfors regular sets in the Heisenberg group characterizes containment in 1-Ahlfors regular curves, with negative results for alpha in (0,2) and a bounded operator on a purely 1-unrectifiable set.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The 1-Ahlfors regularity of the set E together with the non-negativity and precise homogeneity of the kernel K_4 are assumed to control the maximal function and cancellation properties needed for the implication to hold.","pith_extraction_headline":"L^2 boundedness of the K_4 singular integral on a 1-Ahlfors regular set in the Heisenberg group implies it lies in a 1-Ahlfors regular curve."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17680/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.430957Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:11:05.116198Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T21:51:58.768459Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"shingle_duplication","ran_at":"2026-05-19T21:49:44.473472Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T21:49:44.256680Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.528872Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.440974Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"63fd1e0f2f6bbbab4d79262f06b10bbd98ebf03d868b5c66bcbd6dd6d976de62"},"references":{"count":22,"sample":[{"doi":"","year":2012,"title":"V . 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